A man has three friends. The number of ways he can invite one friend everyday for dinner on six successive nights so that no friend is invited more than three times is a. `640` b. `320` c. `420` d. `510`

Options

A. 360
B. 420
C. 170
D. 510

Answer

D

Explanation

Let x,y,z be the friends and a,b,c denote the case when x is invited a times, y is invited b times and z is invited c times. Now, we have the following possibilities (a,b,c)=(1,2,3) or (2,2,2) or (3,3,0)
Hence, the total number of ways
`=(6!)/(1!2!3!)xx3!+(6!)/(2!2!2!)xx(3!)/(3!)+(6!)/(3!3!0!)xx(3!)/(2!)`
`=360+90+60=510`